Question

Use an augmented matrix to solve each system of equations.

$$\begin{gathered} x+y=12 \\ y=x-4 \end{gathered}$$

Short answer

The solution of the given system of equations is $\left(8,4\right)$.

Step-by-step solution

Step 1. Write the augmented matrix for the given system of equations.

The given system of equations is:

$$\begin{gathered} x+y=12 \\ y=x-4 \end{gathered}$$

The given system of equations can be written as:

$$\begin{gathered} x+y=12 \\ x-y=4 \end{gathered}$$

Place the coefficients of the equations and the constant terms into the matrix to form the augmented matrix for the given system of equations.

$\left[\begin{array}{ccc}1& 1& 12\\ 1& -1& 4\end{array}\right]$

Step 2. Solve the given system of equations.

The augmented matrix is:

$\left[\begin{array}{ccc}1& 1& 12\\ 1& -1& 4\end{array}\right]$

Apply $R_2\to R_2-R_1$,

$\left[\begin{array}{ccc}1& 1& 12\\ 0& -2& -8\end{array}\right]$

Apply $R_2\to \frac{-1}{2}{R}_2$,

$\left[\begin{array}{ccc}1& 1& 12\\ 0& 1& 4\end{array}\right]$

Apply $R_1\to R_1-R_2$,

$\left[\begin{array}{ccc}1& 0& 8\\ 0& 1& 4\end{array}\right]$

Therefore, the solution of the given system of equations is $\left(8,4\right)$.